1. Key Concepts Overview

  1. An eigenvector \(x\) lies along the same line as \(Ax\): \[Ax = \lambda x\] The scalar \(\lambda\) is the eigenvalue.

  2. Linear Transformation Properties: If \(Ax = \lambda x\), then:

    • \(A^2x = \lambda^2x\)
    • \(A^{-1}x = \lambda^{-1}x\)
    • \((A + cI)x = (\lambda + c)x\)

  3. Singularity: If \(Ax = \lambda x\), then \((A - \lambda I)x = 0\). This implies the matrix \((A - \lambda I)\) is singular, meaning: \[\det(A - \lambda I) = 0\]

  4. The Two “Quick Checks”:

    • Determinant: \(\det(A) = \lambda_1 \lambda_2 \dots \lambda_n\) (The product of eigenvalues).
    • Trace: \(\text{sum of diagonal entries} = \sum_{i=1}^{n} \lambda_i\) (The sum of eigenvalues).

  5. Special Matrices:

    • Projections: \(\lambda = 1\) and \(0\).
    • Reflections: \(\lambda = 1\) and \(-1\).
    • Rotations: \(\lambda = e^{i\theta}\) and \(e^{-i\theta}\) (Complex values).

2. The Basic Equation

Almost all vectors change direction when multiplied by a matrix \(A\). However, certain exceptional vectors \(x\) stay in the same direction. These are the eigenvectors.

\[Ax = \lambda x\]

The eigenvalue \(\lambda\) tells us whether the special vector \(x\) is stretched, shrunk, reversed, or left unchanged.

Example 1: A 2x2 Matrix

Consider the matrix: \[A = \begin{pmatrix} 0.8 & 0.3 \\ 0.2 & 0.7 \end{pmatrix}\]

To find the eigenvalues, we solve the characteristic equation \(\det(A - \lambda I) = 0\): \[\det \begin{pmatrix} 0.8 - \lambda & 0.3 \\ 0.2 & 0.7 - \lambda \end{pmatrix} = \lambda^2 - 1.5\lambda + 0.5 = 0\]

Factoring the quadratic: \((\lambda - 1)(\lambda - 0.5) = 0\). The eigenvalues are \(\lambda_1 = 1\) and \(\lambda_2 = 0.5\).

3. R Implementation

We can use the built-in eigen() function in R to verify these results.

# Define the matrix A
A <- matrix(c(0.8, 0.2, 0.3, 0.7), nrow = 2, ncol = 2)

# Calculate eigenvalues and eigenvectors
results <- eigen(A)

# Output results
cat("Eigenvalues:\n")
## Eigenvalues:
print(results$values)
## [1] 1.0 0.5
cat("\nEigenvectors (as columns):\n")
## 
## Eigenvectors (as columns):
print(results$vectors)
##           [,1]       [,2]
## [1,] 0.8320503 -0.7071068
## [2,] 0.5547002  0.7071068

4. Geometric Interpretation

4.1 Projection Matrices (\(P\))

A projection matrix projects any vector onto a subspace (e.g., a line or a plane). * If \(x\) is in the column space, it doesn’t move: \(Px = x\) (\(\lambda = 1\)). * If \(x\) is in the nullspace, it is crushed to zero: \(Px = 0\) (\(\lambda = 0\)).

4.2 Reflection Matrices (\(R\))

A reflection matrix flips vectors across a line. * Eigenvectors sitting on the reflection line have \(\lambda = 1\). * Eigenvectors perpendicular to the line are reversed: \(\lambda = -1\).

5. Determinant and Trace

These properties serve as vital checks for manual calculations:

  1. Trace: For matrix \(A\), the trace is \(0.8 + 0.7 = 1.5\). Our eigenvalues \(1 + 0.5 = 1.5\). Check!
  2. Determinant: \(\det(A) = (0.8)(0.7) - (0.3)(0.2) = 0.5\). Our eigenvalues \(1 \times 0.5 = 0.5\). Check!

6. The Characteristic Equation

To find eigenvalues for any \(n \times n\) matrix: 1. Subtract \(\lambda\) from the diagonal: Form the matrix \((A - \lambda I)\). 2. Compute the Determinant: Find the polynomial \(\det(A - \lambda I)\). 3. Find Roots: Solve the polynomial for zero.

Example: Finding Eigenvectors

For \(A = \begin{pmatrix} 1 & 2 \\ 2 & 4 \end{pmatrix}\), \(\det(A - \lambda I) = \lambda^2 - 5\lambda = 0\). Thus \(\lambda_1 = 0\) and \(\lambda_2 = 5\).

  • For \(\lambda = 0\): Solve \((A - 0I)x = 0\). The rows of \(A\) are multiples of \((1, 2)\). The vector \(x_1 = \begin{pmatrix} 2 \\ -1 \end{pmatrix}\) is in the nullspace.
  • For \(\lambda = 5\): Solve \((A - 5I)x = 0\). The vector \(x_2 = \begin{pmatrix} 1 \\ 2 \end{pmatrix}\) stays in its own direction.

7. Imaginary Eigenvalues

Eigenvalues are not always real. Consider a \(90^\circ\) rotation matrix \(Q\): \[Q = \begin{pmatrix} 0 & -1 \\ 1 & 0 \end{pmatrix}\]

The characteristic equation is \(\lambda^2 + 1 = 0\), giving eigenvalues \(\lambda = i\) and \(\lambda = -i\). Since no real vector stays in the same direction after a 90-degree rotation, the eigenvectors must be complex.

8. Real-Life Application: Markov Matrices

A Markov Matrix has columns that sum to 1 (like our Example 1). * Steady State: There is always an eigenvalue \(\lambda = 1\). This represents the “long-term” behavior where the system stabilizes. * Decaying Mode: Other eigenvalues satisfy \(|\lambda| < 1\). * Application: Google’s PageRank algorithm uses the eigenvector of a massive Markov matrix to determine website importance.

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Key Changes Made:

  1. Fixed LaTeX Environments: Changed [ .8 .3 ] to \begin{pmatrix} 0.8 & 0.3 \\ 0.2 & 0.7 \end{pmatrix} for standard mathematical notation.
  2. Corrected Symbols: Replaced Unicode characters like 𝑥 and 𝜆 with standard LaTeX $x$ and $\lambda$.
  3. Improved Logical Flow: Grouped the properties (Trace/Determinant) and the calculation steps (Characteristic Equation) more clearly to match the textbook’s pedagogical style.
  4. Verified Calculations: Added the explicit check for the Trace and Determinant of the Example 1 matrix.